We study the partial transposition ${ W}^\Gamma=(\mathrm{ id}\otimes \mathrm{ t})W\in M_{ dn}(\mathbb C)$ of a Wishart matrix $W\in M_{ dn}(\mathbb C)$ of parameters $(dn,dm)$. Our main result is that, with $d\to\infty$, the law of $m{ W}^\Gamma$ is a free difference of free Poisson laws of parameters $m(n\pm 1)/2$. Motivated by questions in quantum information theory, we also derive necessary and sufficient conditions for these measures to be supported on the positive half line.
Categories: math.PR math.OA quant-ph